Estimated ratios of dark matter and dark energy (which may be the cosmological constant[clarification needed])
in the universe. According to current theories of physics, dark energy
now dominates as the largest source of energy of the universe, in
contrast to earlier epochs when it was insignificant.

In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: Λ) is the value of the energy density of the vacuum of space. It was originally introduced by Albert Einstein in 1917[1] as an addition to his theory of general relativity to "hold back gravity" and achieve a static universe, which was the accepted view at the time. Einstein abandoned the concept after Hubble's 1929 discovery that all galaxies outside the Local Group (the group that contains the Milky Way Galaxy)
are moving away from each other, implying an overall expanding
universe. From 1929 until the early 1990s, most cosmology researchers
assumed the cosmological constant to be zero.

Since the 1990s, several developments in observational cosmology, especially the discovery of the accelerating universe from distant supernovae in 1998 (in addition to independent evidence from the cosmic microwave background and large galaxy redshift surveys), have shown that around 68% of the mass–energy density of the universe can be attributed to dark energy.[2]
While dark energy is poorly understood at a fundamental level, the main
required properties of dark energy are that it functions as a type of
anti-gravity, it dilutes much more slowly than matter as the universe
expands, and it clusters much more weakly than matter, or perhaps not at
all. The cosmological constant is the simplest possible form of dark
energy since it is constant in both space and time, and this leads to
the current standard model of cosmology known as the Lambda-CDM model, which provides a good fit to many cosmological observations.

where the Ricci tensor/scalar R and the metric tensor g describe the structure of spacetime, the stress-energy tensor T describes the energy and momentum density and flux of the matter in that point in spacetime, and the universal constants G and c are conversion factors that arise from using traditional units of measurement. When Λ is zero, this reduces to the field equation of general relativity usually used in the mid-20th century. When T is zero, the field equation describes empty space (the vacuum).

The cosmological constant has the same effect as an intrinsic energy density of the vacuum, ρvac (and an associated pressure). In this context, it is commonly moved onto the right-hand side of the equation, and defined with a proportionality factor of 8π: Λ = 8πρvac, where unit conventions of general relativity are used (otherwise factors of G and c would also appear, i.e. Λ = 8π(G/c2)ρvac = κρvac, where κ is Einstein's constant). It is common to quote values of energy density directly, though still using the name "cosmological constant", with convention8πG = 1. The true dimension of Λ is a length−2.

Given the Planck (2015) values of ΩΛ = 0.6911±0.0062 and Ho = 67.74±0.46 (km/s)/Mpc = (2.195±0.015)×10−18 s−1, Λ has the value of

or 2.90×10−122 in reduced Planck units or 4.33×10−66 eV2 in natural units.

A positive vacuum energy density resulting from a cosmological
constant implies a negative pressure, and vice versa. If the energy
density is positive, the associated negative pressure will drive an
accelerated expansion of the universe, as observed. (See dark energy and cosmic inflation for details.)

ΩΛ (Omega Lambda)

Instead
of the cosmological constant itself, cosmologists often refer to the
ratio between the energy density due to the cosmological constant and
the critical density
of the universe, the tipping point for a sufficient density to stop the
universe from expanding forever. This ratio is usually denoted ΩΛ, and is estimated to be 0.6911±0.0062, according to results published by the Planck Collaboration in 2015.[3]

In a flat universe ΩΛ is the fraction of the energy of the
universe due to the cosmological constant, i.e., what we would
intuitively call the fraction of the universe that is made up of dark
energy. Note that this value changes over time: the critical density
changes with cosmological time,
but the energy density due to the cosmological constant remains
unchanged throughout the history of the universe: the amount of dark
energy increases as the universe grows, while the amount of matter does
not.[citation needed]

Equation of state

Another ratio that is used by scientists is the equation of state, usually denoted w, which is the ratio of pressure that dark energy puts on the universe to the energy per unit volume.[4] This ratio is w = −1 for a true cosmological constant, and is generally different for alternative time-varying forms of vacuum energy such as quintessence.

History

Einstein included the cosmological constant as a term in his field equations for general relativity because he was dissatisfied that otherwise his equations did not allow, apparently, for a static universe:
gravity would cause a universe that was initially at dynamic
equilibrium to contract. To counteract this possibility, Einstein added
the cosmological constant.[5] However, soon after Einstein developed his static theory, observations by Edwin Hubble indicated that the universe appears to be expanding; this was consistent with a cosmological solution to the original general relativity equations that had been found by the mathematician Friedmann,
working on the Einstein equations of general relativity. Einstein later
reputedly referred to his failure to accept the validation of his
equations—when they had predicted the expansion of the universe in
theory, before it was demonstrated in observation of the cosmological red shift—as the "biggest blunder" of his life.[6][dubious– discuss][7]

In fact, adding the cosmological constant to Einstein's equations does not lead to a static universe at equilibrium because the equilibrium is unstable: if the universe expands slightly, then the expansion releases vacuum energy, which causes yet more expansion. Likewise, a universe that contracts slightly will continue contracting.[8]:59

However, the cosmological constant remained a subject of theoretical
and empirical interest. Empirically, the onslaught of cosmological data
in the past decades strongly suggests that our universe has a positive
cosmological constant.[5] The explanation of this small but positive value is an outstanding theoretical challenge (see the section below).

Positive value

Observations announced in 1998 of distance–redshift relation for Type Ia supernovae[9][10] indicated that the expansion of the universe is accelerating. When combined with measurements of the cosmic microwave background radiation these implied a value of ΩΛ ≈ 0.7,[11] a result which has been supported and refined by more recent measurements.[12] There are other possible causes of an accelerating universe, such as quintessence, but the cosmological constant is in most respects the simplest solution. Thus, the current standard model of cosmology, the Lambda-CDM model, includes the cosmological constant, which is measured to be on the order of 10−52 m−2, in metric units. It is often expressed as 10−35 s−2 or 10−122[13] in other unit systems. The value is based on recent measurements of vacuum energy density, ρvacuum=5.96×10−27 kg/m3{\displaystyle \rho _{\text{vacuum}}=5.96\times 10^{-27}{\text{ kg/m}}^{3}},[14] or 10−47 GeV4, 10−29 g/cm3 in other unit systems.

As was only recently seen, by works of 't Hooft, Susskind[15] and others, a positive cosmological constant has surprising consequences, such as a finite maximum entropy of the observable universe.

Predictions

Quantum field theory

A major outstanding problem is that most quantum field theories predict a huge value for the quantum vacuum. A common assumption is that the quantum vacuum
is equivalent to the cosmological constant. Although no theory exists
that supports this assumption, arguments can be made in its favor.[16]
Such arguments are usually based on dimensional analysis and effective field theory. If the universe is described by an effective local quantum field theory down to the Planck scale, then we would expect a cosmological constant of the order of Mpl2{\displaystyle M_{\rm {pl}}^{2}} (6×1054eV2{\displaystyle 6\times 10^{54}\,{\text{eV}}^{2}} in natural unit or 1{\displaystyle 1} in reduced Planck unit). As noted above, the measured cosmological constant is smaller than this by a factor of ~10−120. This discrepancy has been called "the worst theoretical prediction in the history of physics!".[17]

Some supersymmetric theories require a cosmological constant that is exactly zero, which further complicates things. This is the cosmological constant problem, the worst problem of fine-tuning in physics: there is no known natural way to derive the tiny cosmological constant used in cosmology from particle physics.

Anthropic principle

One possible explanation for the small but non-zero value was noted by Steven Weinberg in 1987 following the anthropic principle.[24]
Weinberg explains that if the vacuum energy took different values in
different domains of the universe, then observers would necessarily
measure values similar to that which is observed: the formation of
life-supporting structures would be suppressed in domains where the
vacuum energy is much larger. Specifically, if the vacuum energy is
negative and its absolute value is substantially larger than it appears
to be in the observed universe (say, a factor of 10 larger), holding all
other variables (e.g. matter density) constant, that would mean that
the universe is closed; furthermore, its lifetime would be shorter than
the age of our universe, possibly too short for intelligent life to
form. On the other hand, a universe with a large positive cosmological
constant would expand too fast, preventing galaxy formation. According
to Weinberg, domains where the vacuum energy is compatible with life
would be comparatively rare. Using this argument, Weinberg predicted
that the cosmological constant would have a value of less than a hundred
times the currently accepted value.[25] In 1992, Weinberg refined this prediction of the cosmological constant to 5 to 10 times the matter density.[26]

This argument depends on a lack of a variation of the distribution
(spatial or otherwise) in the vacuum energy density, as would be
expected if dark energy were the cosmological constant. There is no
evidence that the vacuum energy does vary, but it may be the case if,
for example, the vacuum energy is (even in part) the potential of a
scalar field such as the residual inflaton (also see quintessence). Another theoretical approach that deals with the issue is that of multiverse
theories, which predict a large number of "parallel" universes with
different laws of physics and/or values of fundamental constants. Again,
the anthropic principle states that we can only live in one of the
universes that is compatible with some form of intelligent life. Critics
claim that these theories, when used as an explanation for fine-tuning,
commit the inverse gambler's fallacy.

In 1995, Weinberg's argument was refined by Alexander Vilenkin to predict a value for the cosmological constant that was only ten times the matter density,[27] i.e. about three times the current value since determined.

Thursday, March 15, 2018

The Big Bounce is a hypothetical cosmological model for the origin of the known universe. It was originally suggested as a phase of the cyclic model or oscillatory universe interpretation of the Big Bang,
where the first cosmological event was the result of the collapse of a
previous universe. It receded from serious consideration in the early
1980s after inflation theory emerged as a solution to the horizon problem, which had arisen from advances in observations revealing the large-scale structure of the universe. In the early 2000s, inflation was found by some theorists to be problematic and unfalsifiable
in that its various parameters could be adjusted to fit any
observations, so that the properties of the observable universe are a
matter of chance. An alternative picture including a Big Bounce was
conceived as a predictive and falsifiable possible solution to the horizon problem, and has been under active investigation since 2017.[1]

Expansion and contraction

The concept of the Big Bounce envisions the Big Bang as the beginning of a period of expansion that followed a period of contraction. In this view, one could talk of a Big Crunch followed by a Big Bang, or more simply, a Big Bounce.
This suggests that we could be living at any point in an infinite
sequence of universes, or conversely the current universe could be the
very first iteration. However, if the condition of the interval phase
"between bounces", considered the 'hypothesis of the primeval atom', is
taken into full contingency such enumeration may be meaningless because
that condition could represent a singularity in time at each instance, if such perpetual return was absolute and undifferentiated.

The main idea behind the quantum theory of a Big Bounce is that, as density approaches infinity, the behavior of the quantum foam changes. All the so-called fundamental physical constants, including the speed of light in a vacuum, need not remain constant during a Big Crunch, especially in the time interval smaller than that in which measurement may never be possible (one unit of Planck time, roughly 10−43 seconds) spanning or bracketing the point of inflection.

If the fundamental physical constants were determined in a
quantum-mechanical manner during the Big Crunch, then their apparently
inexplicable values in this universe would not be so surprising, it
being understood here that a universe is that which exists between a Big Bang and its Big Crunch.

The Big Bounce Models, however do not explain much about that how the
currently expanding universe will manage to contract. This constant and
steady expansion is explained by NASA through the metric expansion of space.

By the early 1980s, the advancing precision and scope of observational cosmology had revealed that the large-scale structure of the universe is flat, homogenous and isotropic, a finding later accepted as the Cosmological Principle to apply at scales beyond roughly 300 million light-years. It was recognized that it was necessary to find an explanation
for how distant regions of the universe could have essentially
identical properties without ever having been in light-like
communication. A solution was proposed to be a period of exponential
expansion of space in the early universe, as a basis for what became
known as Inflation theory. Following the brief inflationary period, the universe continues to expand, but at a less rapid rate.

Various formulations of inflation theory and their detailed
implications became the subject of intense theoretical study. In the
absence of a compelling alternative, inflation became the leading
solution to the horizon problem. In the early 2000s, inflation was found
by some theorists to be problematic and unfalsifiable in that its
various parameters could be adjusted to fit any observations, a
situation known as a fine-tuning problem. Furthermore, inflation was
found to be inevitably eternal,
creating an infinity of different universes with typically different
properties, so that the properties of the observable universe are a
matter of chance.[3]
An alternative concept including a Big Bounce was conceived as a
predictive and falsifiable possible solution to the horizon problem,[4] and is under active investigation as of 2017.[5][1]

The phrase "Big Bounce" appeared in the scientific literature in
1987, when it was first used in the title of a pair of articles (in
German) in Stern und Weltraum by Wolfgang Priester and Hans-Joachim Blome.[6] It reappeared in 1988 in Iosif Rozental’s Big Bang, Big Bounce,
a revised English-language translation of a Russian-language book (by a
different title), and in a 1991 article (in English) by Priester and
Blome in Astronomy and Astrophysics. (The phrase apparently originated as the title of a novel by Elmore Leonard in 1969, shortly after increased public awareness of the Big Bang model with of the discovery of the cosmic microwave background by Penzias and Wilson in 1965.)

One of the main problems with the Big Bang theory is that at the moment of the Big Bang, there is a singularity
of zero volume and infinite energy. This is normally interpreted as the
end of the physics as we know it; in this case, of the theory of general relativity. This is why one expects quantum effects to become important and avoid the singularity.

However, research in loop quantum cosmology purported to show that a previously existing universe collapsed, not to the point of singularity, but to a point before that where the quantum effects of gravity
become so strongly repulsive that the universe rebounds back out,
forming a new branch. Throughout this collapse and bounce, the evolution
is unitary.

Bojowald also claims that some properties of the universe that
collapsed to form ours can also be determined. Some properties of the
prior universe are not determinable however due to some kind of
uncertainty principle.

This work is still in its early stages and very speculative. Some extensions by further scientists have been published in Physical Review Letters.[8]

In 2003, Peter Lynds
has put forward a new cosmology model in which time is cyclic. In his
theory our Universe will eventually stop expanding and then contract.
Before becoming a singularity, as one would expect from Hawking's black
hole theory, the universe would bounce. Lynds claims that a singularity
would violate the second law of thermodynamics
and this stops the universe from being bounded by singularities. The
Big Crunch would be avoided with a new Big Bang. Lynds suggests the
exact history of the universe would be repeated in each cycle in an eternal recurrence. Some critics argue that while the universe may be cyclic, the histories would all be variants.[citation needed]
Lynds' theory has been dismissed by mainstream physicists for the lack
of a mathematical model behind its philosophical considerations.[9]

In 2011, Nikodem Popławski showed that a nonsingular Big Bounce appears naturally in the Einstein-Cartan-Sciama-Kibble theory of gravity.[11]
This theory extends general relativity by removing a constraint of the
symmetry of the affine connection and regarding its antisymmetric part,
the torsion tensor,
as a dynamical variable. The minimal coupling between torsion and Dirac
spinors generates a spin-spin interaction which is significant in
fermionic matter at extremely high densities. Such an interaction averts
the unphysical Big Bang singularity, replacing it with a cusp-like
bounce at a finite minimum scale factor, before which the universe was
contracting. This scenario also explains why the present Universe at
largest scales appears spatially flat, homogeneous and isotropic,
providing a physical alternative to cosmic inflation.

In 2012, a new theory of nonsingular big bounce was successfully constructed within the frame of standard Einstein gravity.[12] This theory combines the benefits of matter bounce and Ekpyrotic cosmology.
Particularly, the famous BKL instability, that the homogeneous and
isotropic background cosmological solution is unstable to the growth of
anisotropic stress, is resolved in this theory. Moreover, curvature
perturbations seeded in matter contraction are able to form a nearly
scale-invariant primordial power spectrum and thus provides a consistent
mechanism to explain the cosmic microwave background (CMB) observations.

A few sources argue that distant supermassive black holes whose large size is hard to explain so soon after the Big Bang, such as ULAS J1342+0928,[13] may be evidence for a Big Bounce, with these supermassive black holes being formed before the Big Bounce.[14][15]

Given a reference basis of vectors, a tensor can be represented as an organized multidimensional array of numerical values. The order (also degree or rank)
of a tensor is the dimensionality of the array needed to represent it,
or equivalently, the number of indices needed to label a component of
that array. For example, a linear map is represented by a matrix (a
2-dimensional array) in a basis, and therefore is a 2nd-order tensor. A
vector is represented as a 1-dimensional array in a basis, and is a
1st-order tensor. Scalars are single numbers and are thus 0th-order
tensors. The collection of tensors on a vector space forms a tensor algebra.

Because they express a relationship between vectors, tensors
themselves must be independent of a particular choice of basis. The
basis independence of a tensor then takes the form of a covariant and/or contravariant transformation law
that relates the array computed in one basis to that computed in
another one. The precise form of the transformation law determines the type (or valence) of the tensor. The tensor type is a pair of natural numbers (n, m), where n is the number of contravariant indices and m is the number of covariant indices. The total order of a tensor is the sum of these two numbers.

Tensors are important in physics because they provide a concise
mathematical framework for formulating and solving physics problems in
areas such as stress, elasticity, fluid mechanics, and general relativity.
In applications, it is common to study situations in which a different
tensor can occur at each point of an object; for example the stress
within an object may vary from one location to another. This leads to
the concept of a tensor field. In some areas, tensor fields are so ubiquitous that they are simply called "tensors".

Definition

Although
seemingly different, the various approaches to defining tensors
describe the same geometric concept using different languages and at
different levels of abstraction.

As multidimensional arrays

Just as a vector in an n-dimensional space is represented by a one-dimensional array of length n with respect to a given basis, any tensor with respect to a basis is represented by a multidimensional array. For example, a linear operator is represented in a basis as a two-dimensional square n × n array. The numbers in the multidimensional array are known as the scalar components of the tensor or simply its components. They are denoted by indices giving their position in the array, as subscripts and superscripts, following the symbolic name of the tensor. For example, the components of an order 2 tensor T could be denoted Tij , where i and j are indices running from 1 to n, or also by Tji.
Whether an index is displayed as a superscript or subscript depends on
the transformation properties of the tensor, described below. Thus while
Tij and Tji can both be expressed as n by n matrices, and are numerically related via index juggling,
the difference in their transformation laws indicates it would be
improper to add them together. The total number of indices required to
identify each component uniquely is equal to the dimension of the array, and is called the order, degree or rank of the tensor. However, the term "rank" generally has another meaning in the context of matrices and tensors.

Just as the components of a vector change when we change the basis
of the vector space, the components of a tensor also change under such a
transformation. Each type of tensor comes equipped with a transformation law that details how the components of the tensor respond to a change of basis. The components of a vector can respond in two distinct ways to a change of basis (see covariance and contravariance of vectors), where the new basis vectorse^i{\displaystyle \mathbf {\hat {e}} _{i}} are expressed in terms of the old basis vectors ej{\displaystyle \mathbf {e} _{j}} as,

Here Rji are the entries of the change of basis matrix, and in the rightmost expression the summation sign was suppressed: this is the Einstein summation convention, which will be used throughout this article.[Note 1] The components vi of a column vector v transform with the inverse of the matrix R,

where the hat denotes the components in the new basis. This is called a contravariant transformation law, because the vector transforms by the inverse of the change of basis. In contrast, the components, wi, of a covector (or row vector), w transform with the matrix R itself,

w^i=wjRij.{\displaystyle {\hat {w}}_{i}=w_{j}R_{i}^{j}.}

This is called a covariant transformation law, because the covector transforms by the same matrix
as the change of basis matrix. The components of a more general tensor
transform by some combination of covariant and contravariant
transformations, with one transformation law for each index. If the
transformation matrix of an index is the inverse matrix of the basis
transformation, then the index is called contravariant and is
conventionally denoted with an upper index (superscript). If the
transformation matrix of an index is the basis transformation itself,
then the index is called covariant and is denoted with a lower index (subscript).

As a simple example, the matrix of a linear operator with respect to a basis is a rectangular array T{\displaystyle T} that transforms under a change of basis matrix R=(Rij){\displaystyle R=\left(R_{i}^{j}\right)} by T^=R−1TR{\displaystyle {\hat {T}}=R^{-1}TR}. For the individual matrix entries, this transformation law has the form T^j′i′=(R−1)ii′TjiRj′j{\displaystyle {\hat {T}}_{j'}^{i'}=\left(R^{-1}\right)_{i}^{i'}T_{j}^{i}R_{j'}^{j}}
so the tensor corresponding to the matrix of a linear operator has one
covariant and one contravariant index: it is of type (1,1).

Combinations of covariant and contraviant components with the same
index allow us to express geometric invariants. For example, the fact
that a vector is the same object in different coordinate systems can be
captured by the following equations, using the formulas defined above:

where δjk{\displaystyle \delta _{j}^{k}} is the Kronecker delta, which functions similarly to the identity matrix, and has the effect of renaming indices (j into k in this example). This shows several features of the component notation- the ability to re-arrange terms at will (commutativity),
the need to use different indices when working with multiple objects in
the same expression, the ability to rename indices, and the manner in
which contravariant and covariant tensors combine so that all instances
of the transformation matrix and its inverse cancel, so that expressions
like viei{\displaystyle {v}^{i}\,\mathbf {e} _{i}} can immediately be seen to be geometrically identical in all coordinate systems.

Similarly, a linear operator, viewed as a geometric object, does not
actually depend on a basis: it is just a linear map that accepts a
vector as an argument and produces another vector. The transformation
law for the how the matrix of components of a linear operator changes
with the basis is consistent with the transformation law for a
contravariant vector, so that the action of a linear operator on a
contravariant vector is represented in coordinates as the matrix product
of their respective coordinate representations. That is, the components
(Tv)i{\displaystyle (Tv)^{i}} are given by (Tv)i=Tjivj{\displaystyle (Tv)^{i}=T_{j}^{i}v^{j}}. These components transform contravariantly, since

Here the primed indices denote components in the new coordinates, and
the unprimed indices denote the components in the old coordinates. Such
a tensor is said to be of order or type(p, q).
The terms "order", "type", "rank", "valence", and "degree" are all
sometimes used for the same concept. Here, the term "order" or "total
order" will be used for the total dimension of the array (or its
generalisation in other definitions), p + q
in the preceding example, and the term "type" for the pair giving the
number of contravariant and covariant indices. A tensor of type (p, q) is also called a (p, q)-tensor for short.

The definition of a tensor as a multidimensional array satisfying a transformation law traces back to the work of Ricci.[2]

An equivalent definition of a tensor uses the representations of the general linear group. There is an action of the general linear group on the set of all ordered bases of an n-dimensional vector space. If f=(f1,…,fn){\displaystyle \mathbf {f} =(\mathbf {f} _{1},\dots ,\mathbf {f} _{n})} is an ordered basis, and R=(Rji){\displaystyle R=(R_{j}^{i})} is an invertible n×n{\displaystyle n\times n} matrix, then the action is given by

Let F be the set of all ordered bases. Then F is a principal homogeneous space for GL(n). Let W be a vector space and let ρ{\displaystyle \rho } be a representation of GL(n) on W (that is, a group homomorphismρ:GL(n)→GL(W){\displaystyle \rho :{\text{GL}}(n)\to {\text{GL}}(W)}). Then a tensor of type ρ{\displaystyle \rho } is an equivariant mapT:F→W{\displaystyle T:F\to W}. Equivariance here means that

T(FR)=ρ(R−1)T(F).{\displaystyle T(FR)=\rho (R^{-1})T(F).}

When ρ{\displaystyle \rho } is a tensor representation
of the general linear group, this gives the usual definition of tensors
as multidimensional arrays. This definition is often used to describe
tensors on manifolds,[5] and readily generalizes to other groups.[3]

As multilinear maps

A
downside to the definition of a tensor using the multidimensional array
approach is that it is not apparent from the definition that the
defined object is indeed basis independent, as is expected from an
intrinsically geometric object. Although it is possible to show that
transformation laws indeed ensure independence from the basis, sometimes
a more intrinsic definition is preferred. One approach that is common
in differential geometry is to define tensors relative to a fixed (finite-dimensional) vector space V, which is usually taken to be a particular vector space of some geometrical significance like the tangent space to a manifold.[6] In this approach, a type (p, q) tensor T is defined as a multilinear map,

where V∗ is the corresponding dual space of covectors, which is linear in each of its arguments. The above assumes V is a vector space over the real numbers, R. More generally, V can be taken over an arbitrary field of numbers, F (e.g. the complex numbers) with a one-dimensional vector space over F replacing R as the codomain of the multilinear maps.

By applying a multilinear map T of type (p, q) to a basis {ej} for V and a canonical cobasis {εi} for V∗,

a (p + q)-dimensional array of components can be obtained. A different choice of basis will yield different components. But, because T
is linear in all of its arguments, the components satisfy the tensor
transformation law used in the multilinear array definition. The
multidimensional array of components of T thus form a tensor
according to that definition. Moreover, such an array can be realized as
the components of some multilinear map T. This motivates viewing multilinear maps as the intrinsic objects underlying tensors.

In viewing a tensor as a multilinear map, it is conventional to identify the vector space V with the space of linear functionals on the dual of V, the double dualV∗∗. There is always a natural linear map from V to its double dual, given by evaluating a linear form in V∗ against a vector in V. This linear mapping is an isomorphism in finite dimensions, and it is often then expedient to identify V with its double dual.

Using tensor products

For some mathematical applications, a more abstract approach is
sometimes useful. This can be achieved by defining tensors in terms of
elements of tensor products of vector spaces, which in turn are defined through a universal property. A type (p, q) tensor is defined in this context as an element of the tensor product of vector spaces,[7][8]

A basis vi of V and basis wj of W naturally induce a basis vi ⊗ wj of the tensor product V ⊗ W. The components of a tensor T are the coefficients of the tensor with respect to the basis obtained from a basis {ei} for V and its dual basis {εj}, i.e.

Using the properties of the tensor product, it can be shown that these components satisfy the transformation law for a type (p, q) tensor. Moreover, the universal property of the tensor product gives a 1-to-1 correspondence between tensors defined in this way and tensors defined as multilinear maps.

Tensor products can be defined in great generality – for example, involving arbitrary modules
over a ring. In principle, one could define a "tensor" simply to be an
element of any tensor product. However, the mathematics literature
usually reserves the term tensor for an element of a tensor product of any number of copies of a single vector space V and its dual, as above.

Tensors in infinite dimensions

This
discussion of tensors so far assumes finite dimensionality of the
spaces involved, where the spaces of tensors obtained by each of these
constructions are naturally isomorphic.[Note 2] Constructions of spaces of tensors based on the tensor product and
multilinear mappings can be generalized, essentially without
modification, to vector bundles or coherent sheaves.[9]
For infinite-dimensional vector spaces, inequivalent topologies lead to
inequivalent notions of tensor, and these various isomorphisms may or
may not hold depending on what exactly is meant by a tensor (see topological tensor product). In some applications, it is the tensor product of Hilbert spaces
that is intended, whose properties are the most similar to the
finite-dimensional case. A more modern view is that it is the tensors'
structure as a symmetric monoidal category that encodes their most important properties, rather than the specific models of those categories.[10]

Tensor fields

In many applications, especially in differential geometry and
physics, it is natural to consider a tensor with components that are
functions of the point in a space. This was the setting of Ricci's
original work. In modern mathematical terminology such an object is
called a tensor field, often referred to simply as a tensor.[2]
In this context, a coordinate basis is often chosen for the tangent vector space. The transformation law may then be expressed in terms of partial derivatives of the coordinate functions,

Examples

This table shows important examples of tensors on vector spaces and
tensor fields on manifolds. The tensors are classified according to
their type (n, m), where n is the number of contravariant indices, m is the number of covariant indices, and n + m gives the total order of the tensor. For example, a bilinear form is the same thing as a (0, 2)-tensor; an inner product is an example of a (0, 2)-tensor, but not all (0, 2)-tensors are inner products. In the (0, M)-entry of the table, M
denotes the dimensionality of the underlying vector space or manifold
because for each dimension of the space, a separate index is needed to
select that dimension to get a maximally covariant antisymmetric tensor.

Raising an index on an (n, m)-tensor produces an (n + 1, m − 1)-tensor;
this corresponds to moving diagonally down and to the left on the
table. Symmetrically, lowering an index corresponds to moving diagonally
up and to the right on the table. Contraction of an upper with a lower index of an (n, m)-tensor produces an (n − 1, m − 1)-tensor; this corresponds to moving diagonally up and to the left on the table.

Orientation defined by an ordered set of vectors.

Reversed orientation corresponds to negating the exterior product.

Geometric interpretation of grade n elements in a real exterior algebra for n = 0 (signed point), 1 (directed line segment, or vector), 2 (oriented plane element), 3 (oriented volume). The exterior product of n vectors can be visualized as any n-dimensional shape (e.g. n-parallelotope, n-ellipsoid); with magnitude (hypervolume), and orientation defined by that on its n − 1-dimensional boundary and on which side the interior is.[12][13]

Notation

There are several notational systems that are used to describe tensors and perform calculations involving them.

Einstein summation convention

The Einstein summation convention dispenses with writing summation signs, leaving the summation implicit. Any repeated index symbol is summed over: if the index i is used twice in a given term of a tensor expression, it means that the term is to be summed for all i. Several distinct pairs of indices may be summed this way.

Penrose graphical notation

Penrose graphical notation
is a diagrammatic notation which replaces the symbols for tensors with
shapes, and their indices by lines and curves. It is independent of
basis elements, and requires no symbols for the indices.

Abstract index notation

The abstract index notation is a way to write tensors such that the indices are no longer thought of as numerical, but rather are indeterminates. This notation captures the expressiveness of indices and the basis-independence of index-free notation.

Component-free notation

Operations

There
are several operations on tensors that again produce a tensor. The
linear nature of tensor implies that two tensors of the same type may be
added together, and that tensors may be multiplied by a scalar with
results analogous to the scaling of a vector.
On components, these operations are simply performed component-wise.
These operations do not change the type of the tensor; but there are
also operations that produce a tensor of different type.

Tensor product

The tensor product takes two tensors, S and T, and produces a new tensor, S ⊗ T,
whose order is the sum of the orders of the original tensors. When
described as multilinear maps, the tensor product simply multiplies the
two tensors, i.e.

If S is of type (l, k) and T is of type (n, m), then the tensor product S ⊗ T has type (l + n, k + m).

Contraction

Tensor contraction is an operation that reduces a type (n, m) tensor to a type (n − 1, m − 1) tensor, of which the trace
is a special case. It thereby reduces the total order of a tensor by
two. The operation is achieved by summing components for which one
specified contravariant index is the same as one specified covariant
index to produce a new component. Components for which those two indices
are different are discarded. For example, a (1, 1)-tensor Tij{\displaystyle T_{i}^{j}} can be contracted to a scalar through

Tii{\displaystyle T_{i}^{i}}.

Where the summation is again implied. When the (1, 1)-tensor is interpreted as a linear map, this operation is known as the trace.

The contraction is often used in conjunction with the tensor product to contract an index from each tensor.

The contraction can also be understood using the definition of a
tensor as an element of a tensor product of copies of the space V with the space V∗ by first decomposing the tensor into a linear combination of simple tensors, and then applying a factor from V∗ to a factor from V. For example, a tensor

In a vector space with an inner product (also known as a metric) g, the term contraction
is used for removing two contravariant or two covariant indices by
forming a trace with the metric tensor or its inverse. For example, a (2, 0)-tensor Tij{\displaystyle T^{ij}} can be contracted to a scalar through

Tijgij{\displaystyle T^{ij}g_{ij}}

(yet again assuming the summation convention).

Raising or lowering an index

When a vector space is equipped with a nondegenerate bilinear form (or metric tensor
as it is often called in this context), operations can be defined that
convert a contravariant (upper) index into a covariant (lower) index and
vice versa. A metric tensor is a (symmetric) (0, 2)-tensor;
it is thus possible to contract an upper index of a tensor with one of
the lower indices of the metric tensor in the product. This produces a
new tensor with the same index structure as the previous tensor, but
with lower index generally shown in the same position of the contracted
upper index. This operation is quite graphically known as lowering an index.
Conversely, the inverse operation can be defined, and is called raising an index. This is equivalent to a similar contraction on the product with a (2, 0)-tensor. This inverse metric tensor has components that are the matrix inverse of those of the metric tensor.

Applications

Continuum mechanics

Important examples are provided by continuum mechanics. The stresses inside a solid body or fluid are described by a tensor field. The stress tensor and strain tensor are both second-order tensor fields, and are related in a general linear elastic material by a fourth-order elasticity tensor fields.
In detail, the tensor quantifying stress in a 3-dimensional solid
object has components that can be conveniently represented as a 3 × 3
array. The three faces of a cube-shaped infinitesimal volume segment of
the solid are each subject to some given force. The force's vector
components are also three in number. Thus, 3 × 3, or 9 components are
required to describe the stress at this cube-shaped infinitesimal
segment. Within the bounds of this solid is a whole mass of varying
stress quantities, each requiring 9 quantities to describe. Thus, a
second-order tensor is needed.

If a particular surface element
inside the material is singled out, the material on one side of the
surface will apply a force on the other side. In general, this force
will not be orthogonal to the surface, but it will depend on the
orientation of the surface in a linear manner. This is described by a
tensor of type (2, 0), in linear elasticity, or more precisely by a tensor field of type (2, 0), since the stresses may vary from point to point.

Applications of tensors of order > 2

The
concept of a tensor of order two is often conflated with that of a
matrix. Tensors of higher order do however capture ideas important in
science and engineering, as has been shown successively in numerous
areas as they develop. This happens, for instance, in the field of computer vision, with the trifocal tensor generalizing the fundamental matrix.

The field of nonlinear optics studies the changes to material polarization density under extreme electric fields. The polarization waves generated are related to the generating electric fields through the nonlinear susceptibility tensor. If the polarization P is not linearly proportional to the electric field E, the medium is termed nonlinear. To a good approximation (for sufficiently weak fields, assuming no permanent dipole moments are present), P is given by a Taylor series in E whose coefficients are the nonlinear susceptibilities:

Here χ(1){\displaystyle \chi ^{(1)}} is the linear susceptibility, χ(2){\displaystyle \chi ^{(2)}} gives the Pockels effect and second harmonic generation, and χ(3){\displaystyle \chi ^{(3)}} gives the Kerr effect. This expansion shows the way higher-order tensors arise naturally in the subject matter.

Generalizations

Tensor products of vector spaces

The vector spaces of a tensor product
need not be the same, and sometimes the elements of such a more general
tensor product are called "tensors". For example, an element of the
tensor product space V ⊗ W is a second-order "tensor" in this more general sense,[14] and an order-d tensor may likewise be defined as an element of a tensor product of d different vector spaces.[15] A type (n, m) tensor, in the sense defined previously, is also a tensor of order n + m in this more general sense. The concept of tensor product can be extended to arbitrary modules over a ring.

Tensor densities

Suppose that a homogeneous medium fills R3, so that the density of the medium is described by a single scalar value ρ in kg m−3. The mass, in kg, of a region Ω is obtained by multiplying ρ by the volume of the region Ω, or equivalently integrating the constant ρ over the region:

m=∫Ωρdxdydz{\displaystyle m=\int _{\Omega }\rho \,dx\,dy\,dz}

where the Cartesian coordinates xyz
are measured in m. If the units of length are changed into cm, then the
numerical values of the coordinate functions must be rescaled by a
factor of 100:

The numerical value of the density ρ must then also transform by 100−3m3/cm3{\displaystyle 100^{-3}m^{3}/cm^{3}} to compensate, so that the numerical value of the mass in kg is still given by integral of ρdxdydz{\displaystyle \rho \,dx\,dy\,dz}. Thus ρ′=100−3ρ{\displaystyle \rho '=100^{-3}\rho } (in units of kg cm−3).

More generally, if the Cartesian coordinates xyz undergo a linear transformation, then the numerical value of the density ρ must change by a factor of the reciprocal of the absolute value of the determinant of the coordinate transformation, so that the integral remains invariant, by the change of variables formula
for integration. Such a quantity that scales by the reciprocal of the
absolute value of the determinant of the coordinate transition map is
called a scalar density. To model a non-constant density, ρ is a function of the variables xyz (a scalar field), and under a curvilinear change of coordinates, it transforms by the reciprocal of the Jacobian of the coordinate change. For more on the intrinsic meaning, see Density on a manifold.

A tensor density transforms like a tensor under a coordinate change,
except that it in addition picks up a factor of the absolute value of
the determinant of the coordinate transition:[19]

Here w is called the weight. In general, any tensor multiplied
by a power of this function or its absolute value is called a tensor
density, or a weighted tensor.[20][21] An example of a tensor density is the current density of electromagnetism.

Under an affine transformation of the coordinates, a tensor
transforms by the linear part of the transformation itself (or its
inverse) on each index. These come from the rational representations
of the general linear group. But this is not quite the most general
linear transformation law that such an object may have: tensor densities
are non-rational, but are still semisimple
representations. A further class of transformations come from the
logarithmic representation of the general linear group, a reducible but
not semisimple representation,[22] consisting of an (x,y) ∈ R2 with the transformation law

Geometric objects

The transformation law for a tensor behaves as a functor
on the category of admissible coordinate systems, under general linear
transformations (or, other transformations within some class, such as local diffeomorphisms.)
This makes a tensor a special case of a geometrical object, in the
technical sense that it is a function of the coordinate system
transforming functorially under coordinate changes.[23] Examples of objects obeying more general kinds of transformation laws are jets and, more generally still, natural bundles.[24][25]

Spinors

When changing from one orthonormal basis (called a frame)
to another by a rotation, the components of a tensor transform by that
same rotation. This transformation does not depend on the path taken
through the space of frames. However, the space of frames is not simply connected (see orientation entanglement and plate trick):
there are continuous paths in the space of frames with the same
beginning and ending configurations that are not deformable one into the
other. It is possible to attach an additional discrete invariant to
each frame that incorporates this path dependence, and which turns out
(locally) to have values of ±1.[26] A spinor
is an object that transforms like a tensor under rotations in the
frame, apart from a possible sign that is determined by the value of
this discrete invariant.[27][28]
Succinctly, spinors are elements of the spin representation of the rotation group, while tensors are elements of its tensor representations. Other classical groups
have tensor representations, and so also tensors that are compatible
with the group, but all non-compact classical groups have
infinite-dimensional unitary representations as well.

History

The concepts of later tensor analysis arose from the work of Carl Friedrich Gauss in differential geometry, and the formulation was much influenced by the theory of algebraic forms and invariants developed during the middle of the nineteenth century.[29] The word "tensor" itself was introduced in 1846 by William Rowan Hamilton[30] to describe something different from what is now meant by a tensor.[Note 3] The contemporary usage was introduced by Woldemar Voigt in 1898.[31]

Tensor calculus was developed around 1890 by Gregorio Ricci-Curbastro under the title absolute differential calculus, and originally presented by Ricci in 1892.[32] It was made accessible to many mathematicians by the publication of Ricci and Tullio Levi-Civita's 1900 classic text Méthodes de calcul différentiel absolu et leurs applications (Methods of absolute differential calculus and their applications).[33]

In the 20th century, the subject came to be known as tensor analysis, and achieved broader acceptance with the introduction of Einstein's theory of general relativity,
around 1915. General relativity is formulated completely in the
language of tensors. Einstein had learned about them, with great
difficulty, from the geometer Marcel Grossmann.[34]
Levi-Civita then initiated a correspondence with Einstein to correct
mistakes Einstein had made in his use of tensor analysis. The
correspondence lasted 1915–17, and was characterized by mutual respect:

I admire the elegance of your method of computation; it must be nice
to ride through these fields upon the horse of true mathematics while
the like of us have to make our way laboriously on foot.

About Me

My formal training is in chemistry. I also read a great deal of physics and biology. In fact I very much enjoy reading in general, mostly science, but also some fiction and history. I also enjoy computer programming and writing. I like hiking and exploring nature. I also enjoy people; not too much in social settings, but one on one; also, people with interesting or "off-beat" minds draw me to them. I also have some interest in Buddhism.

These days I get a lot more information from the internet, primarily through Wiki. Some television, e. g., documentaries, PBS shows like "Nova" and "Nature".

My favorite science writers are Jacob Bronowski ("The Ascent of Man") and Richard Dawkins (his "The Blind Watchmaker" is right up there up Ascent). I also have a favorite writer on Buddhism, Pema Chodron. Favorite films are "Annie Hall" (by Woody Allen), "The Maltese Falcon", "One Flew Over The Cuckoo's Nest", "As Good As It Gets", "Conspiracy Theory", Monty Python's "Search For The Holy Grail" and "Life of Brian", and a few others which I can't think about at the moment.

I love a number of classical works (Beethoven's "Pastoral", "Afternoon Of A Fawn" and "Clair De Lune" by Debussey , Pachelbel's "Canon" come to mind. My favorite piece is probably Gershwin's "Rhapsody in Blue". But I also enjoy a great deal in modern music, including many jazz pieces, folk songs by people like Dylan, Simon and Garfunkel, a hodgepodge of pieces by Crosby, Stills, and Nash, Niel Young, and practically everything the Beatles wrote.

My life over the last few years has been in some disarray, but I am finally "getting it together.". As I am very much into the sciences and writing, I would like to move more in this direction. I also enjoy teaching. As for my political leanings, most people would probably describe as basically liberal, though not extremely so. My religious leanings are to the absolutely none: I've alluded to my interest in Buddhism, but again this is not any supernatural or scientifically untested aspect of it but in the way it provides a powerful philosophy and set of practical, day to day methods of dealing with myself and the other human beings.